There are two concepts in the Halacha of Bittul (nullification). One says that if there is an item that is Unkosher that falls into another item that is Kosher, and it is a mixture, and the Unkosher is one-sixtieth, it is nullified and you can eat the mixture even though it has the Unkosher in it. However, there is another Halacha along the same premise that says if Roiv (a majority) of the food is Kosher, then the Unkosher food is nullified. Wouldn't you know that from the fact that it is more than SIXTY?

3 Answers
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Min BeMino - a mixture of the same types of food. In this case the prohibited substance (the 'issur') is batel (nullified) if it is in the minority (rov). The rabbis enacted a restriction (gezeira) not to eat the mixture until the issur is less than one 60th of the total (shishim).

Min BeSheEino Mino - a mixture of different types of food. Here, the issur is batel if it no longer imparts flavor (noten taam). To determine this, one would have to have an expert chef (kefeila) taste the food to determine if the flavor is there. Since this is not always practical (and we assume we are not experts at 'tasting') we wait until shishim and then assume the flavor is gone, unless it's a very flavorful food (tavlin) when we usually assume you are just stuck.

(There are way too many sources, but see the Taz YD 98:3 for a similar, succinct, and good enough summary.)

Please make sure to CYLOR before relying on these principles yourself, as there are many small details that can make a difference.

When do we say 'one in two is nullified' according to the Torah? Only
when the forbidden item is by itself and the permitted item is by
itself... since the prohibited item is not discernible it is nullified
by the majority. But when the permitted item acquires flavor from the
prohibited item the entire thing becomes prohibited because it IS
discernible and known throughout the entire permitted item...

The key point is that having one part nullified by the majority is
only possible when the one part is not currently discernible. Since
there is flavor, the whole concept falls apart.

The point of having 60:1 is generally just to guarantee there isn't any significant, discernible flavor, as is clear in YD 98:1 and other places.

Sometimes rov is enough and other times we require shishim. See this chart for details but basically the only case l'halacha we will rely on rov is if no taste is transferred (yavesh byavesh) and taste the same (min bmino).